House Democrats are renewing their push for a vote on a proposed comprehensive immigration reform package, vowing Tuesday to refocus efforts on pressuring Republicans to sign onto a discharge petition that would force a vote on the legislation.

The immigration reform push is the third recent attempt by Republicans to leverage a discharge petition — a procedural tactic that allows the majority of House members to supersede the will of the House leadership and bring a bill to the floor — in an attempt to force a vote on a piece of legislation that they support.

House Democrats say they currently have 191 signatures — all Democrats — on the petition, and that they will recommit to pressure Republican lawmakers who have said previously that they would support comprehensive immigration reform. The petition must get 218 signatures to force a vote on the legislation.

Immigration reform has emerged as the next Republican battleground after a high-profile split between party leaders and the Tea Party over budget compromise led to growing expectation of a similar bipartisan deal to legalise America’s 11 million undocumented migrants.

House speaker John Boehner signalled a major break from right-wing conservatives on Thursday when he accused activist groups of losing “all credibility” by opposing his efforts to reach a deal with Democrats over the $1tn federal budget. Heritage Action, a group behind many Tea Party Republicans, issued a testy response on Friday, claiming Boehner was trying to clear the way for immigration reform next year by severing his links with opponents on the right of the party.

Sen. Charles E. Schumer, New York Democrat, said Thursday that he’d bet “quite a bit” that comprehensive immigration reforming will pass Congress and that he wouldn’t rule out something happening by the end of the year.

Rep. Jeff Denham (R-Calif.), who was the first GOP House member to sign on to a Democrat-led comprehensive immigration reform bill, said Thursday that the effort shouldn’t be declared dead yet.

In fact, he said, three additional members from both parties will announce their support for that bill in the next couple of days, with more to come next week. At the same time, he and five other Republicans are talking to fellow GOP members to get 40 to 45 more to sign a letter supporting immigration reform in general.

In a meeting with business leaders to discuss immigration reform, President Obama predicted that there are enough votes in the House to pass the contentious issue.

“Although right now there’s been some resistance from House Republicans, what’s been encouraging is that there are a number of House Republicans who have said we think this is the right thing to do as well,” Mr. Obama said Tuesday at the White House. “It’s my estimation that we actually have the votes to get comprehensive immigration reform done in the House right now. The politics are challenging for [Speaker Boehner] and others, and we want to make it as easy for them as possible. This is not an issue where we’re looking for a political win, this is one where we’re looking for a substantive win for the U.S. economy and the American people and the businesses that are represented here.”

Minority Leader Nancy Pelosi said Tuesday there is a bipartisan majority in the House of Representatives ready to pass a comprehensive immigration overhaul bill, but she may never get the chance to prove it if GOP Speaker John A. Boehner doesn’t bring a vote in the Republican-lead House.

“With 28 Republicans having publicly expressed support for a path to citizenship, we believe the votes are there on a bipartisan basis to pass a bill,” she wrote on Facebook. “It’s just a question whether Speaker Boehner can muster the will to schedule a vote.”